We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let K be an algebraically closed field of positive characteristic, let G be a finitely generated subgroup of the multiplicative group of K, and let X be an (irreducible) quasiprojective variety defined over K. We consider K-valued sequences of the form a_n:=f(ϕⁿ(x₀)), where ϕ: X-> X and f: X->P¹ are rational maps defined over K and x₀ ∈ X is a point whose forward orbit avoids the indeterminacy loci of ϕ and f. We show that the set of n for which a_n ∈ G is a finite union of arithmetic progressions along with a set of Banach density zero.

We thank the anonymous referee for many helpful comments and suggestions which improved our paper. Both authors were partially supported by NSERC Discovery grants.